Affiliation:
1. Departamento de Geometría y Topología , Universidad de Granada , E-18071 Granada , Spain
Abstract
Abstract
In the present paper we consider the Carnot–Carathéodory distance
δ
E
{\delta_{E}}
to a closed set E in the sub-Riemannian Heisenberg groups
ℍ
n
{{\mathbb{H}}^{n}}
,
n
⩾
1
{n\geqslant 1}
. The
ℍ
{{\mathbb{H}}}
-regularity of
δ
E
{\delta_{E}}
is proved under mild conditions involving a general notion of singular points. In case E is a Euclidean
C
k
{C^{k}}
submanifold,
k
⩾
2
{k\geqslant 2}
, we prove that
δ
E
{\delta_{E}}
is
C
k
{C^{k}}
out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of E is of class
C
2
{C^{2}}
are obtained, out of the singular set, in terms of the horizontal principal curvatures of
∂
E
{\partial E}
and of the function
〈
N
,
T
〉
/
|
N
h
|
{\langle N,T\rangle/|N_{h}|}
and its tangent derivatives.
Subject
Applied Mathematics,Analysis
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